How fast does linear consensus converge on nonreversible graphs?
ANALYSIS OF LINEAR CONSENSUS ALGORITHM ON STRONGLY CONNECTED GRAPH USING EFFECTIVE RESISTANCE
This paper analyzes the performance of the linear consensus algorithm, a fundamental algorithm in multi-agent systems, using a metric called the LQ cost. It extends previous work by analyzing non-reversible consensus networks, which are relevant for directed communication graphs, unlike prior work that focused on reversible cases. The paper introduces the concepts of "back-and-forth paths" and "pivot nodes" to analyze these non-reversible networks, leveraging the notion of effective resistance from electrical network theory.
While not directly addressing LLMs, the analysis of efficient communication and consensus in directed networks is highly relevant to LLM-based multi-agent systems where agents may have asymmetric communication patterns (e.g., one agent querying another, but not vice-versa). The theoretical bounds and analysis provided could inform the design and optimization of communication strategies in such systems, potentially improving efficiency and convergence speed in complex multi-agent interactions. Specifically, understanding the performance implications of non-reversibility is crucial for realistic multi-agent scenarios often encountered with LLMs.